The discrete subgroupMore precisely, the relevant groups are the projective special linear groups PSL(2, \(\mathbb {R}\) ) =
SL(2,\(\mathbb {R}/\lbrace \pm \rbrace \) ) and PSL(2,\(\mathbb {Z}\) ) = SL(2,\(\mathbb {Z})/\lbrace \pm 1\rbrace \) .
\(\mathrm {SL}(2,\mathbb {Z}) =\left\lbrace  \gamma = \left(\begin{array}{cc}a & b\\c & d\end{array}\right)\;\big |\: a, b, c, d \in \mathbb {Z},\, ad-bc=1\right\rbrace \)  of the special linear group \(\mathrm {SL}(2,\mathbb {R})\) , is the symmetry
group of lattices \(\Lambda \)  in the complex plane \(\mathbb {C}\)  [1]}, [2]}, [3]}, [4]}. It is sometimes called
the full modular group, and has the following congruence subgroups of finite indices
\(\Gamma _0(N) &= \left\lbrace \gamma \in \mathrm {SL}(2,\mathbb {Z})\, |\, c \equiv 0\: \text{mod}\: N\right\rbrace \nonumber \\\Gamma _1(N) &= \left\lbrace \gamma \in \mathrm {SL}(2,\mathbb {Z})\, |\, a,d \equiv 1\:\text{and}\: c \equiv 0\: \text{mod}\: N\right\rbrace \\\Gamma (N) &= \left\lbrace \gamma \in \mathrm {SL}(2,\mathbb {Z})\, |\, a,d \equiv 1\:\text{and}\: b,c \equiv 0\: \text{mod}\: N\right\rbrace \nonumber \) 
